Linear Cryptanalysis Using Multiple Linear Approximations

In this article, the theory of multidimensional linear attacks on block ciphers is developed and the basic attack algorithms and their complexity estimates are presented. As an application the multidimensional linear distinguisher derived by Cho for the block cipher PRESENT is discussed in detail

Multidimensional linear cryptanalysis

Linear cryptanalysis is an important tool for studying the security of symmetric ciphers. In 1993 Matsui proposed two algorithms, called Algorithm 1 and Algorithm 2, for recovering information about the secret key of a block cipher. The algorithms exploit a biased probabilistic relation between the input and output of the cipher. This relation is called the (one-dimensional) linear approximation of the cipher. Mathematically, the problem of key recovery is a binary hypothesis testing problem that can be solved with appropriate statistical tools. The same mathematical tools can be used for realising a distinguishing attack against a stream cipher. The distinguisher outputs whether the given sequence of keystream bits is derived from a cipher or a random source. Sometimes, it is even possible to recover a part of the initial state of the LFSR used in a key stream generator. Several authors considered using many one-dimensional linear approximations simultaneously in a key recovery attack and various solutions have been proposed. In this thesis a unified methodology for using multiple linear approximations in distinguishing and key recovery attacks is presented. This methodology, which we call multidimensional linear cryptanalysis, allows removing unnecessary and restrictive assumptions. We model the key recovery problems mathematically as hypothesis testing problems and show how to use standard statistical tools for solving them. We also show how the data complexity of linear cryptanalysis on stream ciphers and block ciphers can be reduced by using multiple approximations. We use well-known mathematical theory for comparing different statistical methods for solving the key recovery problems. We also test the theory in practice with reduced round Serpent. Based on our results, we give recommendations on how multidimensional linear cryptanalysis should be used

Multivariate Profiling of Hulls for Linear Cryptanalysis

Extensions of linear cryptanalysis making use of multiple approximations, such as multiple and multidimensional linear cryptanalysis, are an important tool in symmetric-key cryptanalysis, among others being responsible for the best known attacks on ciphers such as Serpent and present. At CRYPTO 2015, Huang et al. provided a refined analysis of the key-dependent capacity leading to a refined key equivalence hypothesis, however at the cost of additional assumptions. Their analysis was extended by Blondeau and Nyberg to also cover an updated wrong key randomization hypothesis, using similar assumptions. However, a recent result by Nyberg shows the equivalence of linear dependence and statistical dependence of linear approximations, which essentially invalidates a crucial assumption on which all these multidimensional models are based. In this paper, we develop a model for linear cryptanalysis using multiple linearly independent approximations which takes key-dependence into account and complies with Nyberg’s result. Our model considers an arbitrary multivariate joint distribution of the correlations, and in particular avoids any assumptions regarding normality. The analysis of this distribution is then tailored to concrete ciphers in a practically feasible way by combining a signal/noise decomposition approach for the linear hulls with a profiling of the actual multivariate distribution of the signal correlations for a large number of keys, thereby entirely avoiding assumptions regarding the shape of this distribution. As an application of our model, we provide an attack on 26 rounds of present which is faster and requires less data than previous attacks, while using more realistic assumptions and far fewer approximations. We successfully extend the attack to present the first 27-round attack which takes key-dependence into account

Linear Cryptanalysis of DES with Asymmetries

Linear cryptanalysis of DES, proposed by Matsui in 1993, has had a seminal impact on symmetric-key cryptography, having seen massive research efforts over the past two decades. It has spawned many variants, including multidimensional and zero-correlation linear cryptanalysis. These variants can claim best attacks on several ciphers, including PRESENT, Serpent, and CLEFIA. For DES, none of these variants have improved upon Matsui\u27s original linear cryptanalysis, which has been the best known-plaintext key-recovery attack on the cipher ever since. In a revisit, Junod concluded that when using $2^{43}$ known plaintexts, this attack has a complexity of $2^{41}$ DES evaluations. His analysis relies on the standard assumptions of right-key equivalence and wrong-key randomisation. In this paper, we first investigate the validity of these fundamental assumptions when applied to DES. For the right key, we observe that strong linear approximations of DES have more than just one dominant trail and, thus, that the right keys are in fact inequivalent with respect to linear correlation. We therefore develop a new right-key model using Gaussian mixtures for approximations with several dominant trails. For the wrong key, we observe that the correlation of a strong approximation after the partial decryption with a wrong key still shows much non-randomness. To remedy this, we propose a novel wrong-key model that expresses the wrong-key linear correlation using a version of DES with more rounds. We extend the two models to the general case of multiple approximations, propose a likelihood-ratio classifier based on this generalisation, and show that it performs better than the classical Bayesian classifier. On the practical side, we find that the distributions of right-key correlations for multiple linear approximations of DES exhibit exploitable asymmetries. In particular, not all sign combinations in the correlation values are possible. This results in our improved multiple linear attack on DES using 4 linear approximations at a time. The lowest computational complexity of $2^{38.86}$ DES evaluations is achieved when using $2^{42.78}$ known plaintexts. Alternatively, using $2^{41}$ plaintexts results in a computational complexity of $2^{49.75}$ DES evaluations. We perform practical experiments to confirm our model. To our knowledge, this is the best attack on DES

New Results in the Linear Cryptanalysis of DES

Two open problems on using Matsui\u27s Algorithm 2 with multiple linear approximations posed earlier by Biryukov, De Canni$\grave{\hbox{e}}$re and M. Quisquater at Crypto\u2704 are solved in the present paper. That improves the linear cryptanalysis of 16-round DES reported by Matsui at Crypto\u2794

Multidimensional Zero-Correlation Linear Cryptanalysis of the Block Cipher KASUMI

The block cipher KASUMI is widely used for security in many synchronous wireless standards. It was proposed by ETSI SAGE for usage in 3GPP (3rd Generation Partnership Project) ciphering algorthms in 2001. There are a great deal of cryptanalytic results on KASUMI, however, its security evaluation against the recent zero-correlation linear attacks is still lacking so far. In this paper, we select some special input masks to refine the general 5-round zero-correlation linear approximations combining with some observations on the $FL$ functions and then propose the 6-round zero-correlation linear attack on KASUMI. Moreover, zero-correlation linear attacks on the last 7-round KASUMI are also introduced under some weak keys conditions. These weak keys take $2^{-14}$ of the whole key space. The new zero-correlation linear attack on the 6-round needs about $2^{85}$ encryptions with $2^{62.8}$ known plaintexts. For the attack under weak keys conditions on the last 7 round, the data complexity is about $2^{62.1}$ known plaintexts and the time complexity $2^{110.5}$ encryptions

Improved Integral and Zero-correlation Linear Cryptanalysis of Reduced-round CLEFIA Block Cipher

CLEFIA is a block cipher developed by Sony Corporation in 2007. It is a recommended cipher of CRYPTREC, and has been adopted as ISO/IEC international standard in lightweight cryptography. In this paper, some new 9-round zero-correlation linear distinguishers of CLEFIA are constructed with the input masks and output masks being independent, which allow multiple zero-correlation linear attacks on 14/15-rounds CLEAIA-192/256 with the partial sum technique. Furthermore, the relations between integral distinguishers and zero-correlation linear approximations are improved, and some new integral distinguishers over 9-round are deduced from zero-correlation linear approximations. By using these integral distinguishers and the partial sum technique, the previous integral results on CLEFIA are improved. The two results have either one more rounds or lower time complexity than previous attack results by means of integral and zero-correlation linear cryptanalysis

Functional Cryptanalysis: Application to reduced-round Xoodoo

This paper proposes functional cryptanalysis, a flexible and versatile approach to analyse symmetric-key primitives with two primary features. Firstly, it is a generalization of multiple attacks including (but not limited to) differential, rotational and rotational-xor cryptanalysis. Secondly, it is a theoretical framework that unifies all of the aforementioned cryptanalysis techniques and at the same time opens up possibilities for the development of new cryptanalytic approaches. The main idea of functional cryptanalysis is the usage of binary relations in the form of functions, hence the name functional, instead of binary operations like in a classical settings of differential -like cryptanalysis. We establish the theoretical foundations of functional cryptanalysis from standard terminologies. This work also presents an interpretation of functional cryptanalysis from the point of view of commutative algebra. In particular, we exhibit an algorithm to compute the functional probability (hence differential, rotational, and rotational-xor probability) using Grobner bases. We demonstrate the applicability of functional cryptanalysis against reduced-round Xoodoo and compare it against the best differential. To avoid dealing with invalid differential trails, we propose a method to construct a valid differential trail using Satisfiability Modulo Theory (SMT). To the best of our knowledge, this is the first time the SMT model is used to construct a valid differential while previous approaches rely on Mixed-Integer Linear Programming (MILP) model. Lastly, we remark that the use of non-translation functionals shares analogous advantages and limitations with the use of nonlinear approximations in linear cryptanalysis